Is the Big Rip Unreachable?

Physics Letters B 785 (2018) 132–135 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Is the Big Rip unreachable? Konstantinos Dimopoulos Consortium for Fundamental Physics, Physics Department, Lancaster University, Lancaster LA1 4YB, UK a r t i c l e i n f o a b s t r a c t Article history: I investigate the repercussions of particle production when the Universe is dominated by a hypothetical Received 15 July 2018 phantom substance. I show that backreaction due to particle production prevents the density from Received in revised form 2 August 2018 shooting to infinity at a Big Rip, but instead forces it to stabilise at a large constant value. Afterwards Accepted 3 August 2018 there is a period of de-Sitter inflation. I speculate that this might lead to a cyclic Universe. Available online 24 August 2018 © 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license Editor: M. Trodden (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. A phantom substance is defined as a fluid which violates the perature the Hawking temperature of de Sitter space T = H/2π null energy condition because its pressure is p < −ρ, where ρ [1–7]1 is its density. Such a hypothetical substance is also called exotic 2 4 matter, and it is necessary for keeping a wormhole traversable. π gr H ρgr = q g∗ , (1) In cosmology, if the Universe content is dominated by phantom 30 2π density, then the latter increases and at some finite time tmax it gr becomes infinite, in a singularity called the Big Rip. This is be- where g∗ is the number of effective relativistic degrees of free- cause the Universe undergoes super-inflation where the acceler- dom which undergo particle production (i.e. are light and non- ∼ ated expansion gives rise to an event horizon, whose dimensions conformally invariant) and q 1is some constant factor due to shrink to zero at tmax, ripping apart all structures, from galaxies to the fact that the resulting ρgr is not actually thermal. atoms. Thus, the Friedmann equation obtains an additional term of the But is it so? So far, most of the studies of the dynamics of form the Universe when undergoing accelerated expansion have ignored 2 8π G 4 particle production due to the event horizon, because it is usually H = ρ + CH , (2) 3 negligible, in inflation for example. In this work, it is argued that particle production cannot be ignored when the accelerated ex- where ρ is the density of the substance which causes the acceler- pansion is driven by phantom density. Backreaction due to particle ated expansion and, in view of Eq. (1), we have production renders the Big Rip singularity unreachable. 4 8π G qG gr We work in Einstein gravity only. In the following, we use CH = ρgr ⇒ C = g∗ . (3) natural units, where c = h¯ = 1 and Newton’s gravitational con- 3 180π = −2 = × 18 stant is 8π G mP , with mP 2.43 10 GeV being the reduced Strictly speaking, the above are true for ρ, H constant, which Planck mass. For simplicity, isotropy and spatial flatness is as- results in quasi-de Sitter inflation. In this case, the CH4 contri- sumed throughout. bution in the Friedmann equation (2)is negligible and is usually The existence of an event horizon during accelerated expan- ignored (but see Ref. [9]). However, one expects the same phe- sion results in particle production of all light (meaning with mass nomenon to take place in any kind of accelerated expansion, since smaller than the Hubble rate m < H) non-conformally invariant gravitational particle production occurs whenever there is an event horizon (as with black hole radiation, for example). Thus, qualita- fields (e.g. a light scalar field, such as the inflaton). During quasi- 4 tively, the same source term CH would appear in the Friedmann de Sitter inflation, particle production results in the gravitational equation as H sets the scale of the accelerated expansion and generation of density of the order of thermal density with tem- 1 Note that, particle production is a non-equilibrium effect and in an isotropic E-mail address: [email protected]. universe its classical counterpart is bulk viscosity [8]. https://doi.org/10.1016/j.physletb.2018.08.040 0370-2693/© 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. K. Dimopoulos / Physics Letters B 785 (2018) 132–135 133 √ √ √ gr 2 C ∝ g∗ as in Eq. (3). This term would become important if the 1 2 + 2CH 2 − 1 1 3 √ ln √ √ √ − 2 1 − √ cause of accelerated expansion is a phantom substance. 2 2 − 2CH 2 + 1 2CH If ρ is the density of a phantom substance then the Fried- | + | mann equation (ignoring the extra term CH4, because it can be = 3 1√ w − ˙ (tmax t), (7) negligible at first) results in ρ˙ > 0 and H > 0. This results in super- 2C inflation which leads to the Big Rip singularity when ρ →∞ in where t corresponds to the time when = . Note that finite time. However, the growth of H means that the CH4 in the max ρ ρmax Friedmann equation (2)willeventually become important, because √Eq. (4)(with √the negative sign in the brackets) implies that 1/2 2 2CH= (1 − 1 − u) , where we have defined u ≡ ρ/ρmax, i.e. it increases faster than the H term. This may halt the growth of u(tmax) = 1. ρ and prevent the Big Rip from happening [12,13]. √ When H Hmax = 1/ 2C, the last term in the left-hand-side The solution to Eq. (2)is −1 3 of the above dominates and we find H = |1 + w|(tmax − t) 2 as with standard phantom dark energy, only the time tmax does 1 32π GC ∞ H2 = 1 ± 1 − ρ . (4) not denote the Big Rip, i.e. ρ(tmax) , but instead we have 2C 3 ρ(tmax) = ρmax which is finite and given by Eq. (5). The solution in Eq. (7)only applies for t ≤ tmax. At tmax the 2 For small density, the above equation results in either H = 1/C = 6 solution becomes zero. We can investigate what happens when 2 constant or H = (8π G/3)ρ, which is the usual Friedmann equa- t > tmax by considering the continuity equation (6), which gives tion. The latter solution corresponds to the negative sign in the brackets. However, the above also shows that there is a maximum 3|1 + w| u˙ = 3|1 + w|Hu ⇒ u˙ = √ > 0 , (8) possible value of ρ, which is ρmax = 3/32π GC, in √which case there max = 2C is a maximum value of the Hubble rate Hmax 1/ 2C. This means √ that the Big Rip is unreachable, prevented by the backreaction of where u˙ max ≡ u˙(tmax) and we used that Hmax = 1/ 2C and 4 the gravitational particle production. u(tmax) = 1. Thus, there is a tendency for the density to be- Using Eq. (3), we can obtain an estimate of the maximum den- come larger than ρmax, which, however, is forbidden by Eq. (4). sity This is really because Eq. (2)is not valid any more. Indeed, for ρ > ρmax, the density of the gravitationally produced particles 2 would be larger than the phantom density itself, which cannot = 3 = 135 = 3Hmax ρmax gr happen (where is the energy coming from?). 32π GC 8qG2 g∗ 16π G This suggests that the continuity equation needs augmenting, √ 1/4 ⇒ 1/4 = 30 since the removal of energy by gravitational particle production is ρmax 6π gr mP . (5) qg∗ not considered in Eq. (8); it is implicitly assumed negligible. To take this into account we introduce a negative source term on the 1/4 gr Thus, ρmax can be smaller than the Planck scale only when g∗ is right-hand-side of Eq. (6), which becomes (see also Ref. [15]) 1/4 very large. Considering string theory, we can reduce ρmax to the 17 gr 5 9|1 + w|C string scale ∼ 10 GeV by considering g∗ ∼ 10 or so. We assume 5 ρ˙ − 3|1 + w|Hρ =−3|1 + w|Hρgr =− H . (9) 1/4 8π G that ρmax < mP such that quantum gravity considerations can be 5 ignored. 4 The form of this term is given by δρ/δt, where δρ =−ρgr ∝ H − Exactly how the density evolves can be revealed by the study as in Eq. (1)and δt ∼ H 1 is the Hubble time. This is because the of the continuity equation relativistic particles produced gravitationally in a Hubble time are diluted by the accelerated expansion and replenished by the parti- ρ˙ + 3(1 + w)Hρ = 0 , (6) cles produced in the following Hubble time, meaning that δρ ∝ H4 −1 is produced gravitationally per Hubble time δt ∼ H . The precise −1 where w = p/ρ < −1is the barotropic parameter of the phantom value is δt = H /3|1 + w|, which results in u˙ max = 0. substance. For simplicity, we consider w = constant. Using Eq. (4) Combining Eqs. (4) and (9)we may write the continuity equa- with the negative sign in the brackets, Eq. (6)can be analytically tion as solved to give √ √ 3/2 u˙ = 6|1 + w|Hmax 1 − u(1 − 1 − u) , (10) √ 2 = 1 | + = ˙ = In general, in accelerated expansion the Hawking temperature is T 2 3w where Hmax 1/ 2C. From the above it is evident that umax 0 1|(H/2π) [10].

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